Integrand size = 104, antiderivative size = 30 \[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=e^{e^{e^{-3+\frac {5}{x}-2 x+\frac {-\frac {e^5}{x}+x}{e^5}}}} \] Output:
exp(exp(exp((x-exp(5)/x)/exp(5)-2*x+5/x-3)))
Time = 3.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=e^{e^{e^{-3+\frac {4}{x}+\left (-2+\frac {1}{e^5}\right ) x}}} \] Input:
Integrate[(E^(-5 + E^E^((x^2 + E^5*(4 - 3*x - 2*x^2))/(E^5*x)) + E^((x^2 + E^5*(4 - 3*x - 2*x^2))/(E^5*x)) + (x^2 + E^5*(4 - 3*x - 2*x^2))/(E^5*x))* (x^2 + E^5*(-4 - 2*x^2)))/x^2,x]
Output:
E^E^E^(-3 + 4/x + (-2 + E^(-5))*x)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^2+e^5 \left (-2 x^2-4\right )\right ) \exp \left (\frac {x^2+e^5 \left (-2 x^2-3 x+4\right )}{e^5 x}+e^{e^{\frac {x^2+e^5 \left (-2 x^2-3 x+4\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (-2 x^2-3 x+4\right )}{e^5 x}}-5\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \int \frac {\left (\left (1-2 e^5\right ) x^2-4 e^5\right ) \exp \left (\frac {x^2+e^5 \left (-2 x^2-3 x+4\right )}{e^5 x}+e^{e^{\frac {x^2+e^5 \left (-2 x^2-3 x+4\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (-2 x^2-3 x+4\right )}{e^5 x}}-5\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (\left (1-2 e^5\right ) x^2-4 e^5\right ) \exp \left (-\frac {-\left (\left (1-2 e^5\right ) x^2\right )-e^{e^{\left (\frac {1}{e^5}-2\right ) x+\frac {4}{x}-3}+5} x-e^{\left (\frac {1}{e^5}-2\right ) x+\frac {4}{x}+2} x+8 e^5 x-4 e^5}{e^5 x}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\left (1-2 e^5\right ) \exp \left (-\frac {-\left (\left (1-2 e^5\right ) x^2\right )-e^{e^{\left (\frac {1}{e^5}-2\right ) x+\frac {4}{x}-3}+5} x-e^{\left (\frac {1}{e^5}-2\right ) x+\frac {4}{x}+2} x+8 e^5 x-4 e^5}{e^5 x}\right )-\frac {4 \exp \left (5-\frac {-\left (\left (1-2 e^5\right ) x^2\right )-e^{e^{\left (\frac {1}{e^5}-2\right ) x+\frac {4}{x}-3}+5} x-e^{\left (\frac {1}{e^5}-2\right ) x+\frac {4}{x}+2} x+8 e^5 x-4 e^5}{e^5 x}\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (1-2 e^5\right ) \int \exp \left (-\frac {-\left (\left (1-2 e^5\right ) x^2\right )-e^{5+e^{\left (-2+\frac {1}{e^5}\right ) x-3+\frac {4}{x}}} x-e^{\left (-2+\frac {1}{e^5}\right ) x+2+\frac {4}{x}} x+8 e^5 x-4 e^5}{e^5 x}\right )dx-4 \int \frac {\exp \left (-\frac {-\left (\left (1-2 e^5\right ) x^2\right )-e^{5+e^{\left (-2+\frac {1}{e^5}\right ) x-3+\frac {4}{x}}} x-e^{\left (-2+\frac {1}{e^5}\right ) x+2+\frac {4}{x}} x+3 e^5 x-4 e^5}{e^5 x}\right )}{x^2}dx\) |
Input:
Int[(E^(-5 + E^E^((x^2 + E^5*(4 - 3*x - 2*x^2))/(E^5*x)) + E^((x^2 + E^5*( 4 - 3*x - 2*x^2))/(E^5*x)) + (x^2 + E^5*(4 - 3*x - 2*x^2))/(E^5*x))*(x^2 + E^5*(-4 - 2*x^2)))/x^2,x]
Output:
$Aborted
Time = 5.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
| method | result | size |
| norman | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {\left (\left (-2 x^{2}-3 x +4\right ) {\mathrm e}^{5}+x^{2}\right ) {\mathrm e}^{-5}}{x}}}}\) | \(29\) |
| parallelrisch | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {\left (\left (-2 x^{2}-3 x +4\right ) {\mathrm e}^{5}+x^{2}\right ) {\mathrm e}^{-5}}{x}}}}\) | \(29\) |
| risch | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-\frac {\left (2 x^{2} {\mathrm e}^{5}+3 x \,{\mathrm e}^{5}-x^{2}-4 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{x}}}}\) | \(33\) |
Input:
int(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5))*exp( exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5)))*exp(exp(exp(((-2*x^2-3*x+4)*exp (5)+x^2)/x/exp(5))))/x^2/exp(5),x,method=_RETURNVERBOSE)
Output:
exp(exp(exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5))))
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (25) = 50\).
Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.53 \[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=e^{\left (-\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x} + \frac {{\left (x^{2} - 2 \, {\left (x^{2} + 4 \, x - 2\right )} e^{5} + x e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x} + 5\right )} + x e^{\left (e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x}\right )} + 5\right )}\right )} e^{\left (-5\right )}}{x} - e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x}\right )} + 5\right )} \] Input:
integrate(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5) )*exp(exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5)))*exp(exp(exp(((-2*x^2-3*x+ 4)*exp(5)+x^2)/x/exp(5))))/x^2/exp(5),x, algorithm="fricas")
Output:
e^(-(x^2 - (2*x^2 + 3*x - 4)*e^5)*e^(-5)/x + (x^2 - 2*(x^2 + 4*x - 2)*e^5 + x*e^((x^2 - (2*x^2 + 3*x - 4)*e^5)*e^(-5)/x + 5) + x*e^(e^((x^2 - (2*x^2 + 3*x - 4)*e^5)*e^(-5)/x) + 5))*e^(-5)/x - e^((x^2 - (2*x^2 + 3*x - 4)*e^ 5)*e^(-5)/x) + 5)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=e^{e^{e^{\frac {x^{2} + \left (- 2 x^{2} - 3 x + 4\right ) e^{5}}{x e^{5}}}}} \] Input:
integrate(((-2*x**2-4)*exp(5)+x**2)*exp(((-2*x**2-3*x+4)*exp(5)+x**2)/x/ex p(5))*exp(exp(((-2*x**2-3*x+4)*exp(5)+x**2)/x/exp(5)))*exp(exp(exp(((-2*x* *2-3*x+4)*exp(5)+x**2)/x/exp(5))))/x**2/exp(5),x)
Output:
exp(exp(exp((x**2 + (-2*x**2 - 3*x + 4)*exp(5))*exp(-5)/x)))
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=e^{\left (e^{\left (e^{\left (x e^{\left (-5\right )} - 2 \, x + \frac {4}{x} - 3\right )}\right )}\right )} \] Input:
integrate(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5) )*exp(exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5)))*exp(exp(exp(((-2*x^2-3*x+ 4)*exp(5)+x^2)/x/exp(5))))/x^2/exp(5),x, algorithm="maxima")
Output:
e^(e^(e^(x*e^(-5) - 2*x + 4/x - 3)))
\[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (x^{2} - 2 \, {\left (x^{2} + 2\right )} e^{5}\right )} e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x} + e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x}\right )} + e^{\left (e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x}\right )}\right )} - 5\right )}}{x^{2}} \,d x } \] Input:
integrate(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5) )*exp(exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5)))*exp(exp(exp(((-2*x^2-3*x+ 4)*exp(5)+x^2)/x/exp(5))))/x^2/exp(5),x, algorithm="giac")
Output:
integrate((x^2 - 2*(x^2 + 2)*e^5)*e^((x^2 - (2*x^2 + 3*x - 4)*e^5)*e^(-5)/ x + e^((x^2 - (2*x^2 + 3*x - 4)*e^5)*e^(-5)/x) + e^(e^((x^2 - (2*x^2 + 3*x - 4)*e^5)*e^(-5)/x)) - 5)/x^2, x)
Time = 0.89 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx={\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}}}} \] Input:
int(-(exp(exp(-(exp(-5)*(exp(5)*(3*x + 2*x^2 - 4) - x^2))/x))*exp(-5)*exp( exp(exp(-(exp(-5)*(exp(5)*(3*x + 2*x^2 - 4) - x^2))/x)))*exp(-(exp(-5)*(ex p(5)*(3*x + 2*x^2 - 4) - x^2))/x)*(exp(5)*(2*x^2 + 4) - x^2))/x^2,x)
Output:
exp(exp(exp(-2*x)*exp(-3)*exp(4/x)*exp(x*exp(-5))))
\[ \int \frac {e^{-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}} \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx=\int \frac {\left (\left (-2 x^{2}-4\right ) {\mathrm e}^{5}+x^{2}\right ) {\mathrm e}^{\frac {\left (-2 x^{2}-3 x +4\right ) {\mathrm e}^{5}+x^{2}}{x \,{\mathrm e}^{5}}} {\mathrm e}^{{\mathrm e}^{\frac {\left (-2 x^{2}-3 x +4\right ) {\mathrm e}^{5}+x^{2}}{x \,{\mathrm e}^{5}}}} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {\left (-2 x^{2}-3 x +4\right ) {\mathrm e}^{5}+x^{2}}{x \,{\mathrm e}^{5}}}}}}{x^{2} {\mathrm e}^{5}}d x \] Input:
int(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5))*exp( exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5)))*exp(exp(exp(((-2*x^2-3*x+4)*exp (5)+x^2)/x/exp(5))))/x^2/exp(5),x)
Output:
int(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5))*exp( exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5)))*exp(exp(exp(((-2*x^2-3*x+4)*exp (5)+x^2)/x/exp(5))))/x^2/exp(5),x)